The aim of this proposal is to develop practical procedures for the analysis of disease incidence data in genetic epidemiology, allowing for censored observation of outcomes, ascertainment probabilities, measured environmental exposures or covariates, genetic markers, unmeasured genetic and environmental factors, and interactions among them. The underlying model is an extension of "frailty" models for multivariate survival data in which the correlation in outcomes is modeled in terms of one or more latent variables ("frailties") that are shared by family members. In the frailty models that have been developed so far, all members of a family are assumed to have the same frailty. Although this allows one to test the phenomenon of familial aggregation, it precludes exploration of the genetic or environmental basis of such aggregation. In the proposed extension of the method, each individual would have a unique frailty, subject to some known correlational structure induced by family relationships with a small number of parameters to be estimated. Major gene(s) and polygenic models of inheritance will be considered, together with linkage to genetic markers, environmental risk factors, and gene-environment (GxE) interactions. We propose to fit the modal and estimate the posterior distribution of modal parameters using a Monte Carlo technique known as Gibbs sampling, in which each unknown quantity (frailty or modal parameter) is successively replaced by random values drawn from their respective posterior distributions, given the observed data and the current values of the other unknowns. This work will consist of four activities, (1) Further theoretical development of the methods, including major-gene and multifactorial models, adaptations required for case-control and other designs, adjustments for ascertainment and missing data, and asymptotic calculations; (2) Development of practical computer programs, using sequential and parallel processing techniques, as well as simple approximations that allow valid analysis using standard packages; (3) Simulation studies of the performance of the methods compared with standard methods in genetics and epidemiology; this will include assessment of the power to distinguish alternative genetic models and patterns of GxE interactions; and (4) Application to various data sets, including breast cancer in families of probands with premenopausal bilateral breast cancer, Alzheimer disease in extended pedigrees, and cardiovascular disease and breast cancer in twins.